# Geometric Analysis

The basic theme of our research is the study of the analytic, geometric and topological properties of mappings under various analytic assumptions. This often leads to the need to understand the geometries of the underlying spaces. We study modern areas, most of which have their origins in geometric function theory. Mapping problems naturally appear whenever one looks for parametrizations, minimizers, or solutions to analytic equations. Besides analysis, our research is directly connected to several different areas of mathematics, such as differential geometry, partial differential equations, geometric topology, and geometric group theory.

## Members

Academy professor Pekka Koskela

Professor Jani Onninen

Professor Kai Rajala

Professor Mikko Salo

Academy research fellow Pekka Pankka

Academy research fellow Tapio Rajala

Academy research fellow Heli Tuominen

University lecturer Antti Käenmäki

University lecturer Enrico Le Donne

Academy postdoc researcher Juha Lehrbäck

Lecturer Päivi Lammi

**Research interests**

### Analysis on metric spaces

The purpose of this modern area is to develop analysis in non-smooth, or even fractal spaces. New applications to problems where smoothness is not present appear all the time. We study weakly differentiable mappings in singular spaces: Sobolev and BV-functions, Lipschitz and quasiconformal mappings. The Poincare and isoperimetric inequalities as well as heat kernel and other analytic estimates are studied as tools to find out the geometric properties of the underlying spaces. The theory of analysis in metric spaces has shown in particular that surprisingly many deep results in analysis only have little dependence on the structure of the underlying space.

### Geometry of domains

Euclidean domains that admit an extension, say for each p-integrable Sobolev function with p-integrable first order derivatives to the entire Euclidean space can be used as a substitute for the entire space in many questions. Such domains also form natural examples for analysis on metric spaces. Sometimes already the validity of a Poincare type inequality is sufficient. We have studied geometric criteria for extendability of Sobolev functions or for the validity of such inequalities. Many challenging problems remain.

Our research on the Hardy inequalities is also related to the geometry of domains. We study in particular the connections between the validity of these important inequalities and the size and geometry of the boundary of the domain.

### Harmonic maps

### Sobolev mappings and non-linear analysis

Minimizers of suitable energy integrals modeling the physical properties of, say, elastic materials, are sense-preserving Sobolev mappings with some analytic properties. These properties are often familiar in quasiconformal analysis, although weaker. We study the properties of such mappings under minimal regularity assumptions, motivated by the connections to both non-linear elasticity theory and quasiconformal analysis. We try to find the minimal assumptions guaranteeing continuity or partial continuity, invertibility, preservance of sets of measure zero, and regularity of the mappings as well as their inverses; properties which have natural physical interpretations. Similarly, we try to find sharp extensions of the basic results in quasiconformal analysis and geometric function theory.

### Quasiconformal analysis

Quasiconformal and quasiregular mappings extend and generalize conformal maps and complex analytic functions, respectively. In particular, the theory developed for them over the past fifty years extends geometric function theory to higher dimensions. Although the theory is already extensive, there are several fundamental problems that remain unanswered, such as the optimal regularity of mappings, the rigidity of higher dimensional mappings compared to the two-dimensional, boundary behavior, as well as the topological properties. More recently, the theory has been extended to cover mappings between singular spaces, with succesful applications to several problems, for instance in geometric group theory and in the classification of metric spaces.

### Self-similar and related sets

The framework of self-similar sets has provided a sufficiently easy environment to produce highly irregular sets. Self-similar sets are often considered as kind of model-fractals and they are used extensively in many areas of mathematics including dynamical systems and analysis. The self-similar sets provide a natural starting point for various generalizations. One possibility is to look at self-affine sets. As a phenomenon, self-affinity often occurs in nature; a shoreline, lungs, and a fern are common examples of these kind of objects. Furthermore, the dynamical system associated to the self-affine set is sub-additive. This is contrary to the self-similar (and self-conformal) set, where the corresponding dynamical system is additive.

### Local geometry of fractal measures

The geometric properties of fractals and their relation to different notions of dimension have been an object of intensive study for several decades. In the recent years, our research group has made several significant contributions in this area. In particular, questions dealing with porosity and conical densities have been studied thoroughly. Porosity is a quantity that measures the size and abundance of holes. Conical density results are used to derive geometric information from metric information. They are useful because they tell how the measure is distributed in small balls.

### Measure and dimension theory in metric spaces

Along with Euclidean spaces, mathematical analysis is focusing more and more into general metric spaces. As these abstract spaces often have fractal features, it is very important to develop the machinery of geometric measure theory in these spaces in order to get better understanding of the local structures. Many of the research themes discussed above are related to and make sense in quite general metric spaces.

### Optimal mass transportation

Optimal mass transportation has been used in many areas: for instance in probability, economics and meteorology. Inside mathematics it connects to many of the research areas of our Department: Optimal transportation equations are closely related to nonlinear PDEs, for instance, transportation with the quadratic cost function is related to the Monge-Ampere equation. Optimal mass transportation has been used to develop Sobolev-space theory in metric measure spaces and it has also been used to define abstract notions of Ricci-curvature lower bounds in metric measure spaces. It can also be used in the study of rectifiability.

## Publications

**Some recent publications**

- Z. Balogh, P. Koskela, S. Rogovin: Absolute continuity of quasiconformal mappings on curves, Geom. Funct. Anal., 17 (2007), no. 3, 645--664.
- C. Bandt, A. Käenmäki: Local structure of self-affine sets, Ergodic Theory Dynam. Systems, 33 (2013), 1326-1337
- D. Beliaev, E. Järvenpää, M. Järvenpää, A. Käenmäki, T. Rajala, S. Smirnov, V. Suomala: Packing dimension of mean porous measures, J. Lond. Math. Soc. 80 (2009), no. 2, 514-530
- J. Heinonen, P. Pankka, K. Rajala: Quasiconformal frames, Arch. Rational Mech. Anal., 196 (2010), no. 3, 839--866.
- S. Hencl, P. Koskela: Regularity of the inverse of a planar Sobolev homeomorphism, Arch. Rational Mech. Anal., 180 (2006), no. 1, 75--95.
- T. Iwaniec, L. V. Kovalev, J. Onninen: The Nitsche conjecture, J. Amer. Math. Soc., 24 (2011), 345--373.
- T. Iwaniec, N.-T. Koh, L. V. Kovalev, J. Onninen: Existence of energy-minimal diffeomorphisms between doubly connected domains, Invent. Math., 186 (2011), no. 3, 667-707
- T. Iwaniec, J. Onninen: Hyperelastic deformations of smallest total energy, Arch. Rational Mech. Anal., 194 (2009), no. 3, 927--986.
- Piotr Hajasz, Pekka Koskela, Heli Tuominen: Sobolev embeddings, extensions and measure density condition, J. Funct. Anal. 254 (2008), no.5, 1217--1234
- R. Korte, J. Lehrbäck, H. Tuominen: The equivalence between pointwise Hardy inequalities and uniform fatness, Math. Ann., 351 (2011), no. 3, 711-731.
- P. Koskela, D. Yang, Y. Zhou: A characterization of Hajlasz-Sobolev and Triebel-Lizorkin spaces via grand Littlewood-Paley functions, J. Funct. Anal. 258 (2010), no. 8, 2637-2661.
- K. Rajala: Radial limits of quasiregular local homeomorphisms, Amer. J. Math., 130 (2008), no. 1, 269--289.
- T. Rajala: Local Poincaré inequalities from stable curvature conditions on metric spaces, Calc. Var. Partial Differential Equations, 44 (2012), no. 3-4, 477-494.